Question 1 of 43intermediate✨ CreateEssay2 marks

Try and find other patterns like this one. For example, you could consider 4 consecutive squares and see if you can find a pattern.

Correct Answer

Consider four consecutive square numbers, such as 1, 4, 9, and 16. If we add the smallest and largest squares, and then subtract the sum of the two middle squares, we find a consistent pattern.

For 1, 4, 9, 16: (1 + 16) - (4 + 9) = 17 - 13 = 4.

Let's try another set of four consecutive squares: 4, 9, 16, 25. Applying the same rule: (4 + 25) - (9 + 16) = 29 - 25 = 4.

Just like the pattern for three consecutive squares always results in 2, the pattern for four consecutive squares always results in 4.

Exercise: Think and Reflect | Q: 1 | (Page 1)
For More Understanding

Explanation

The textbook introduces a pattern for three consecutive squares where (1st + 3rd) - (2 * 2nd) = 2. Extending this logic to four consecutive squares, we test the difference between the sum of the outer squares and the sum of the inner squares. Algebraically, if the squares are n2n^{2}, (n+1)^2, (n+2)^2, and (n+3)^2, the sum of outer squares is n2n^{2} + (n+3)^2 = 2n22n^{2} + 6n + 9, and the sum of inner squares is (n+1)^2 + (n+2)^2 = 2n22n^{2} + 6n + 5. Their difference is always 4.

Solution Steps

  1. Step 1: Select four consecutive square numbers, e.g., 1, 4, 9, 16.

  2. Step 2: Add the smallest and largest squares: 1 + 16 = 17.

  3. Step 3: Add the two middle squares: 4 + 9 = 13.

  4. Step 4: Subtract the sum of the middle squares from the sum of the outer squares: 17 - 13 = 4.

  5. Step 5: Repeat with another set (4, 9, 16, 25) to confirm: (4 + 25) - (9 + 16) = 29 - 25 = 4. The pattern is always 4.

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